Statistically self-similar fractal sets

Proceedings of the American Mathematical Society, 2005

For self-similar sets, the existence of a feasible open set is a natural separation condition which expresses geometric as well as measure-theoretic properties. We give a constructive approach by defining a central open set and characterizing those points which do not belong to feasible open sets.

2001

We study random recursive constructions with finite "memory" in complete metric spaces and the Hausdorff dimension of the generated random fractals. With each such construction and any positive number β we associate a linear operator V (β) in a finite dimensional space. We prove that under some conditions on the random construction the Hausdorff dimension of the fractal coincides with the value of the parameter β for which the spectral radius of V (β) equals 1.

Journal of Mathematical Analysis and Applications, 1992

131) have considered a class of real functions whose graphs are, in general, fractal sets in R2. In this paper we give sufficient conditions for the fractal and Hausdorff dimensions to be equal for a certain subclass of fractal functions. The sets we consider are examples of self-affine fractals generated using iterated function systems (i.f.s.). Falconer [S] has shown that for almost all such sets the fractal and Hausdorff dimensions are equal and he gives a formula for the common dimension, due originally to Moran [S]. These results, however, give no information about individual fractal functions, In this paper we extend Moran's original method and show that if certain conditions on the i.f.s. are satisfied, then the two dimensions are equal. Kono [ 111 and Bedford [ 121 considered special cases of the subclass of fractal functions that we will introduce. Bedford and Urbanski [13] use a nonlinear setting to present conditions for the equality of Hausdorff and fractal dimension. However, their criteria are based on measure-theoretic characterizations and the use of the concept of generalized pressure. Our criterion on the other hand is based on the underlying geometry of the attractor and is easier to verify. We will show this on two specific examples which are more general than the self-ahine functions presented in [13].

Transactions of the American Mathematical Society, 2014

We investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural ‘dimension pair’. In particular, we compute these dimensions for certain classes of self-affine sets and quasi-self-similar sets and study their relationships with other notions of dimension, such as the Hausdorff dimension for example. We also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity.

In this paper, we use fractal structures to study a new approach to the Hausdorff dimension from both continuous and discrete points of view. We show that it is possible to generalize the Hausdorff dimension in the context of Euclidean spaces equipped with their natural fractal structure. To do this, we provide three definitions of fractal dimension for a fractal structure and study their relationships and mathematical properties. One of these definitions is in terms of finite coverings by elements of the fractal structure. We prove that this dimension is equal to the Hausdorff dimension for compact subsets of Euclidean spaces. This may be the key for the creation of new algorithms to calculate the Hausdorff dimension of these kinds of space.

Ergodic Theory and Dynamical Systems, 2016

We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and Mandelbrot percolation. In each setting we compute either thealmost sureor theBaire typicalAssouad dimension and consider some illustrative examples. Our results reveal a phenomenon common to each of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure-theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.

Physics Letters A, 1985

Several definitions of generahzed fractal dimensions are reviewed, generalized, and interconnected. They concern (i) different ways of averaging when treating fractal measures (instead of sets); (ii) "'partial dimensions" measuring the fraetality in different directions, and adding up to the generalized dimensions discussed before.

Lobachevskii Journal of Mathematics, 2003

The problem on intersection of Cantor sets was examined in many papers. To solve this problem, we introduce the notion of lacunary self-similar set. The main difference to the standard (Hutchinson) notion of self-similarity is that the set of similarities used in the construction may vary from step to step in a certain way. Using a modification of method described in [3], [4], we find the Hausdorff dimension of a lacunary self-similar set.

Lobachevskii Journal of Mathematics, 2003

The problem on intersection of Cantor sets was examined in many papers. To solve this problem, we introduce the notion of lacunary self-similar set. The main difference to the standard (Hutchinson) notion of self-similarity is that the set of similarities used in the construction may vary from step to step in a certain way. Using a modification of method described in [3], [4], we find the Hausdorff dimension of a lacunary self-similar set.

Fractals in Physics, 1986

The fractal dimensions of various types of Jntersection sets of random fractals are discussed. ffris frÀipi a better understanding oÍ'issues concernlng-upper critical dimensionality' F'lory upp"oiimátions or universality oi critical behaviour w'lth respect to different excluded volume màttranisms. Diffusion on sel?-intersecting fractals, and on fractals with local bridges' is also iónjiaereA in the light of the above discussion, and accurate results for the relative exponents' obtained from ana'lysis of Monte-Carlo generated enumeratlons, are presented.